A posteriori error analysis of a virtual element method for stress-assisted diffusion problems

We develop and analyse residual-based a posteriori error estimates for the virtual element discretisation of a nonlinear stress-assisted diffusion problem. The model problem involves a two-way coupling between elasticity and diffusion equations in perturbed saddle-point form. A robust global inf-sup condition {under a small data assumption} and a Helmholtz decomposition for H(div,Ω) lead to a reliable and efficient error estimator based on appropriately weighted norms. The complete analysis is carried out in two dimensions and we provide the required ingredients for the extension to the three dimensional case. The analysis relies on novel quasi-interpolation operators for Stokes and edge virtual element spaces. Since these interpolation operators and the corresponding error estimates are non-trivial in the three-dimensional case and are of independent interest, we provide the detailed proofs, while several remaining arguments follow analogously to the two-dimensional case. Finally, we present numerical experiments in both 2D and 3D to demonstrate the optimal performance of the proposed error estimator.